On a drive from your home to town, you

wish to average 52 mph. The distance from
your home to town is 106 miles. However,
at 53 miles (half way), you find you have
averaged only 39 mph.
What average speed must you maintain in
the remaining distance in order to have an
overall average speed of 52 mph?
Answer in units of mph

Well, well, well, looks like you've got yourself a little speed predicament! Let's do some clown math to figure this out.

At the halfway point, you've managed to only average 39 mph. But fear not, my friend, for I'm here to help!

To make up for that not-so-impressive speed, you'll have to make the second half of your journey count. Here's the plan: You want to average 52 mph over the entire trip, which is a total distance of 106 miles.

Since you've already traveled 53 miles at an average of 39 mph, you've used up a certain amount of time. But don't worry, we're not counting calories here!

To maintain an average speed of 52 mph, you need to figure out how fast you'll have to go for the remaining 53 miles. We can use a simple equation to calculate this:

(total average speed) = (total distance) / (total time)

So, plugging in the numbers we know:

52 mph = 106 miles / (total time)

Now, let's solve for the remaining time, shall we?

(total time) = 106 miles / 52 mph
(total time) ≈ 2.04 hours

Since you've already used some time for the first half at an average of 39 mph, we'll subtract that from the total time:

(remaining time) = (total time) - (time spent on first half)
(remaining time) ≈ 2.04 hours - (53 miles / 39 mph) ≈ 1.36 hours

Phew! We're almost there!

Now, let's calculate the average speed you need to maintain in the remaining 53 miles to achieve an overall average speed of 52 mph:

(remaining average speed) = (remaining distance) / (remaining time)
(remaining average speed) = 53 miles / 1.36 hours
(remaining average speed) ≈ 39.04 mph

There you have it! To have an overall average speed of 52 mph, you'll need to maintain an average speed of approximately 39.04 mph in the remaining distance.

Now, hop back in your car and drive like the wind, my friend! Safe travels, clown style!

To find the average speed you must maintain in the remaining distance, we can use the average speed formula:

Average speed = Total distance / Total time

We know the total distance is 106 miles and we want the average speed to be 52 mph. Let's find the total time spent.

Let d1 be the distance covered until the halfway point, which is 53 miles.
Let t1 be the time taken to cover the first half of the distance at an average speed of 39 mph.

We can calculate t1 as follows:
t1 = d1 / v1
t1 = 53 miles / 39 mph

Now we need to find the remaining distance, d2, and the remaining time, t2.

d2 = total distance - d1
d2 = 106 miles - 53 miles

We want to find the average speed needed to cover the remaining distance in order to achieve an overall average speed of 52 mph. Let's call this speed v2.

t2 = d2 / v2

The total time taken is the sum of t1 and t2.

Total time = t1 + t2

Now we can set up an equation using the average speed formula:

52 mph = total distance / (t1 + t2)

Plugging in the known values:

52 mph = 106 miles / (53 miles / 39 mph + d2 / v2)

Substituting the values of d1 and t1 we found earlier:

52 mph = 106 miles / (53 miles / 39 mph + d2 / v2)
52 mph = 106 miles / (53 miles / 39 mph + (106 miles - 53 miles) / v2)

Simplifying:

52 mph = 106 miles / (53/39 + 53/v2)
52 mph = 106 miles / (1.359 + 53/v2)

Now we can solve for v2.

52 mph = 106 miles / (1.359 + 53/v2)
52 mph * (1.359 + 53/v2) = 106 miles
(1.359 + 53/v2) = 106 miles / 52 mph
1.359 + 53/v2 = 2.0384615384615383 mph

Subtracting 1.359 from both sides:

53/v2 = 2.038 - 1.359
53/v2 = 0.679

Dividing both sides by 53:

1/v2 = 0.679/53
1/v2 = 0.012820

Taking the reciprocal of both sides:

v2 = 1/0.012820
v2 = 78

Therefore, you must maintain an average speed of 78 mph in the remaining distance in order to have an overall average speed of 52 mph.

To find the average speed you need to maintain in the remaining distance, let's first calculate the time it took you to cover the first half of the distance.

Given:
Distance from home to town = 106 miles
Distance covered in the first half = 53 miles
Average speed in the first half = 39 mph

To find the time taken for the first half of the distance, we can use the formula:

Time = Distance / Speed

Time taken for the first half = 53 miles / 39 mph = 1.359 hours

Now, let's find the time you have left to cover the remaining distance.

Total time available for the whole trip = Time taken for the first half = 1.359 hours

To find the remaining time, we subtract the time taken for the first half from the total time available:

Remaining time = Total time - Time taken for the first half
Remaining time = 1.359 hours - 1.359 hours = 0 hours

Since you have zero time remaining for the second half, you need to find the speed required to cover the remaining distance in no time.

The average speed is calculated by dividing the total distance by the total time:

Average Speed = Total Distance / Total Time

Average Speed = (106 miles) / (1.359 hours + 0 hours)

Average Speed = 106 miles / 1.359 hours

Average Speed ≈ 78.01 mph

Therefore, to maintain an overall average speed of 52 mph, you would need to maintain an average speed of approximately 78.01 mph for the remaining distance from half-way to town.

To drive 106 miles at an average speed of 52 mph requires a total driving time of

106/52 = 2.03846 hours.

If you drive the first 53 miles at an average speed of 39 mph, you have used up 53/39 = 1.35897 hours

You still need to cover 53 miles in the remaining time available, 0.67949 hrs.
That means that you have to average
53/0.67949 = 80.000 mph during the last half.